import OAI.Geometry.PinnedDistances.ExtremalCore namespace OAI universe u1 u2 section noncomputable section open NumberField open scoped BigOperators noncomputable section open MeasureTheory open scoped Topology noncomputable section open MeasureTheory open scoped Topology ENNReal noncomputable section open MeasureTheory NumberField open scoped BigOperators noncomputable section open scoped BigOperators open MeasureTheory noncomputable section open Filter open scoped Topology noncomputable section open scoped BigOperators namespace WeakPinned.Core.Fibers open scoped BigOperators variable {V : Type u1} {I : Type u2} [Fintype V] [Fintype I] [DecidableEq V] variable (G : Fibers V I) /-- Retain a specified family of the original fibers, restricted to D. This never alters a source or creates a new distance coincidence. -/ def retained (D : Finset V) (Q : I → Prop) [DecidablePred Q] : Fibers V I where source := G.source targets e := if Q e then G.targets e ∩ D else ∅ no_loop e := by split_ifs · exact fun h => G.no_loop e (Finset.mem_inter.mp h).1 · simp disjoint e f hef hs := by apply Finset.disjoint_left.mpr intro y hy hf by_cases he : Q e <;> by_cases hh : Q f · simp only [he,hh,ite_eq_left] at hy hf exact (Finset.disjoint_left.mp (G.disjoint e f hef hs)) (Finset.mem_inter.mp hy).1 (Finset.mem_inter.mp hf).1 all_goals simp [he,hh] at hy hf lemma retained_card (D : Finset V) (Q : I → Prop) [DecidablePred Q] : (((G.retained D Q).edgeSet (-1) D).card : ℝ) = ∑ e, if Q e ∧ G.source e ∈ D then (G.size D e : ℝ) else 0 := by rw [edgeSet_card] apply Finset.sum_congr rfl intro e _ have hsize : (G.retained D Q).size D e = if Q e then G.size D e else 0 := by by_cases hq : Q e <;> simp [size,retained,hq] have ha : (G.retained D Q).active (-1) D e ↔ G.source e ∈ D := by change (G.source e ∈ D ∧ (-1:ℝ)<((G.retained D Q).size D e : ℝ)) ↔ _ have hh : (-1:ℝ)<((G.retained D Q).size D e : ℝ) := by have hh := Nat.cast_nonneg (α:=ℝ) ((G.retained D Q).size D e); linarith simp [hh] simp only [ha,hsize] by_cases hq : Q e <;> by_cases hs : G.source e ∈ D <;> simp [hq,hs] end WeakPinned.Core.Fibers namespace WeakPinned open scoped BigOperators def heavyFiber (P : Finset Plane) (s K lam : ℝ) (C D : Finset P) (e : fiberIndices P s) : Prop := (planarFibers P s).active K C e ∧ lam*((planarFibers P s).size C e : ℝ) ≤ (planarFibers P s).size D e instance (P : Finset Plane) (s K lam : ℝ) (C D : Finset P) : DecidablePred (heavyFiber P s K lam C D) := Classical.decPred _ lemma heavySubset_edge_rich (P : Finset Plane) (s K lam r : ℝ) (C D : Finset P) (hlam : 0 ≤ lam) (hK : (D.card : ℝ)^r ≤ lam*K) (x y : P) (hxy : ((planarFibers P s).retained D (heavyFiber P s K lam C D)).edge (-1) D x y) : (x.val,y.val) ∈ richPairs (planeSubset P D) r := by obtain ⟨e,ha,hsrc,hy⟩ := hxy have hxD : x ∈ D := hsrc ▸ ha.1 have hyD := (Finset.mem_inter.mp hy).2 have hyT := (Finset.mem_inter.mp hy).1 change y ∈ (if heavyFiber P s K lam C D e then (planarFibers P s).targets e ∩ D else ∅) at hyT split_ifs at hyT with he · obtain ⟨hactive,hheavy⟩ := he have hsz : (D.card : ℝ)^r ≤ ((planarFibers P s).size D e : ℝ) := hK.trans ((mul_le_mul_of_nonneg_left (le_of_lt hactive.2) hlam).trans hheavy) have hyT' := (Finset.mem_inter.mp hyT).1 have hsrc' : e.val.1=x.val := congrArg Subtype.val hsrc have hne : x.val ≠ y.val := by intro hh have hd := (planar_target_iff P s e y).mp hyT' rw [hsrc',hh,dist_self] at hd exact (index_radius_pos P s e).ne hd apply Finset.mem_filter.mpr refine ⟨Finset.mem_product.mpr ⟨Finset.mem_image.mpr ⟨x,hxD,rfl⟩, Finset.mem_image.mpr ⟨y,hyD,rfl⟩⟩,hne,?_⟩ rw [planeSubset_card] have hc := Finset.card_le_card (restricted_targets_subset P s D e y hyT') rw [Finset.card_image_of_injective _ Subtype.val_injective,hsrc'] at hc exact hsz.trans (by exact_mod_cast hc) · simp at hyT /-- The exact finite counting step underlying both uniform subset estimates for the rich-pair graph. Each ordered pair is counted only once. -/ theorem heavySubset_count (P : Finset Plane) (s K lam r : ℝ) (C D : Finset P) (hlam : 0 ≤ lam) (hK : (D.card : ℝ)^r ≤ lam*K) : (∑ e, if heavyFiber P s K lam C D e ∧ (planarFibers P s).source e ∈ D then ((planarFibers P s).size D e : ℝ) else 0) ≤ (richPairs (planeSubset P D) r).card := by rw [← (planarFibers P s).retained_card] suffices h : (((planarFibers P s).retained D (heavyFiber P s K lam C D)).edgeSet (-1) D).card ≤ (richPairs (planeSubset P D) r).card by exact_mod_cast h rw [← Finset.card_image_of_injective _ (subtype_pair_injective P)] apply Finset.card_le_card intro z hz obtain ⟨xy,he,rfl⟩ := Finset.mem_image.mp hz exact heavySubset_edge_rich P s K lam r C D hlam hK xy.1 xy.2 (Finset.mem_filter.mp he).2 end WeakPinned namespace WeakPinned open Filter open scoped Topology def Theta (s : ℝ) : ℝ := limsup (fun n : ℕ => F n s) atTop lemma F_bdd (s : ℝ) : IsBoundedUnder (· ≤ ·) atTop (fun n : ℕ => F n s) := isBoundedUnder_of_eventually_le (Eventually.of_forall (fun n => F_le_one n s)) lemma F_cobdd (s : ℝ) : IsCoboundedUnder (· ≤ ·) atTop (fun n : ℕ => F n s) := isCoboundedUnder_le_of_le atTop (fun n => F_nonneg n s) lemma Theta_nonneg (s : ℝ) : 0 ≤ Theta s := by have hh := limsup_le_limsup (f:=atTop) (u:=fun _ : ℕ => (0:ℝ)) (v:=fun n => F n s) (Eventually.of_forall (fun n => F_nonneg n s)) (isCoboundedUnder_le_of_le atTop (fun _ => le_refl (0:ℝ))) (F_bdd s) change 0 ≤ limsup (fun n : ℕ => F n s) atTop simpa only [limsup_const] using hh lemma Theta_le_one (s : ℝ) : Theta s ≤ 1 := limsup_le_of_le (F_cobdd s) (Eventually.of_forall (fun n => F_le_one n s)) lemma Theta_antitone : Antitone Theta := by intro s t hst exact limsup_le_limsup (Eventually.of_forall (fun n => F_antitone n hst)) (F_cobdd t) (F_bdd s) lemma Theta_eq_zero_of_one_le {s : ℝ} (hs : 1 ≤ s) : Theta s=0 := by simp [Theta,F_eq_zero_of_one_le _ hs] lemma tendsto_F_of_Theta_zero {s : ℝ} (hs : Theta s=0) : Tendsto (fun n : ℕ => F n s) atTop (𝓝 0) := by apply tendsto_order.mpr constructor · intro a ha exact Eventually.of_forall (fun n => ha.trans_le (F_nonneg n s)) · intro a ha exact eventually_lt_of_limsup_lt (by change Theta s < a; simpa [hs] using ha) (F_bdd s) def criticalExponent : ℝ := sSup {s : ℝ | 0 < s ∧ 0 < Theta s} lemma positive_exponents_bdd : BddAbove {s : ℝ | 0 < s ∧ 0 < Theta s} := by refine ⟨1,?_⟩ intro s hs by_contra hn have hz := Theta_eq_zero_of_one_le (le_of_lt (lt_of_not_ge hn)) linarith [hs.2] lemma failed_main_nonempty (h : ¬ MainStatement) : {s : ℝ | 0 < s ∧ 0 < Theta s}.Nonempty := by by_contra hn apply h intro s hs apply tendsto_F_of_Theta_zero have hnp : ¬0 < Theta s := fun hp => hn ⟨s,hs,hp⟩ exact le_antisymm (le_of_not_gt hnp) (Theta_nonneg s) lemma critical_pos (h : ¬ MainStatement) : 0 < criticalExponent := by obtain ⟨s,hs,hθ⟩ := failed_main_nonempty h exact hs.trans_le (le_csSup positive_exponents_bdd ⟨hs,hθ⟩) lemma critical_le_one (h : ¬ MainStatement) : criticalExponent ≤ 1 := csSup_le (failed_main_nonempty h) (fun s hs => by by_contra hn have hz := Theta_eq_zero_of_one_le (le_of_lt (lt_of_not_ge hn)) linarith [hs.2]) lemma Theta_pos_below_critical (h : ¬ MainStatement) {s : ℝ} (hs : s < criticalExponent) : 0 < Theta s := by obtain ⟨t,ht,hst⟩ := exists_lt_of_lt_csSup (failed_main_nonempty h) hs exact ht.2.trans_le (Theta_antitone (le_of_lt hst)) lemma Theta_zero_above_critical (h : ¬ MainStatement) {s : ℝ} (hs : criticalExponent < s) : Theta s=0 := by apply le_antisymm _ (Theta_nonneg s) by_contra hn have hp : 0 < Theta s := lt_of_not_ge hn have hs0 := (critical_pos h).trans hs have hh : s ≤ criticalExponent := le_csSup positive_exponents_bdd ⟨hs0,hp⟩ linarith /-- A fixed exponent regime obtained from failure of the uniform multiplicity limit. -/ theorem extremal_parameters (h : ¬ MainStatement) : ∃ c s γ : ℝ, 0 < c ∧ c < criticalExponent/3 ∧ max c ((1-c)*criticalExponent) < s ∧ s < criticalExponent ∧ c < γ ∧ γ < s ∧ ContinuousAt Theta s ∧ 0 < Theta s := by let c := criticalExponent/4 have hp := critical_pos h have h1 := critical_le_one h have hc0 : 0 < c := by dsimp [c]; linarith have hc3 : c < criticalExponent/3 := by dsimp [c]; linarith have hcm : c < criticalExponent := by dsimp [c]; linarith have hm : (1-c)*criticalExponent < criticalExponent := by nlinarith have hmax : max c ((1-c)*criticalExponent) < criticalExponent := max_lt hcm hm have hd := (Theta_antitone.countable_not_continuousAt).dense_compl ℝ obtain ⟨s,hs,hr⟩ := hd.exists_mem_open isOpen_Ioo (Set.nonempty_Ioo.mpr hmax) have hcs : c < s := (le_max_left _ _).trans_lt hr.1 refine ⟨c,s,(c+s)/2,hc0,hc3,hr.1,hr.2,?_,?_,?_,Theta_pos_below_critical h hr.2⟩ · linarith · linarith · simpa only [Set.mem_compl_iff,Set.mem_ofPred_eq,not_not] using hs lemma extremal_subsequence (s : ℝ) : ∃ N : ℕ → ℕ, Tendsto N atTop atTop ∧ Tendsto (fun i => F (N i) s) atTop (𝓝 (Theta s)) := by obtain ⟨N,hN,hT⟩ := exists_seq_tendsto_limsup (F_cobdd s) (F_bdd s) exact ⟨N,hT,hN⟩ /-- The epsilon-uniform extremal tail bound. The lower cutoff may vary arbitrarily provided it tends to infinity. -/ lemma extremal_tail {s : ℝ} (hs : ContinuousAt Theta s) {r : ℕ → ℝ} {M : ℕ → ℕ} (hr : Tendsto r atTop (𝓝 s)) (hM : Tendsto M atTop atTop) : ∀ ε > 0, ∀ᶠ i in atTop, ∀ m : ℕ, M i ≤ m → F m (r i) ≤ Theta s+ε := by intro ε hε obtain ⟨δ,hδ,hcont⟩ := Metric.continuousAt_iff.mp hs (ε/2) (by linarith) let t := s-δ/2 have ht : t < s := by dsimp [t]; linarith have hdist : dist t s < δ := by rw [Real.dist_eq,abs_of_neg (by dsimp [t]; linarith)] dsimp [t] linarith have hΘ : Theta t < Theta s+ε/2 := by have hh := (abs_lt.mp (show |Theta t-Theta s| < ε/2 from by simpa only [Real.dist_eq] using hcont hdist)).2 linarith have hev : ∀ᶠ m : ℕ in atTop, F m t < Theta s+ε := eventually_lt_of_limsup_lt (lt_trans hΘ (by linarith)) (F_bdd t) obtain ⟨m₀,hm₀⟩ := eventually_atTop.mp hev filter_upwards [hr.eventually_const_lt ht,(tendsto_atTop.mp hM) m₀] with i hi hmi intro m hm exact (F_antitone m (le_of_lt hi)).trans (le_of_lt (hm₀ m (hmi.trans hm))) end WeakPinned namespace WeakPinned open Filter open scoped Topology def cutoff (s : ℝ) (N : ℕ) : ℝ := (N : ℝ)^s/Real.log N lemma eventual_power_below_cutoff {s r : ℝ} (hrs : r < s) : ∀ᶠ N : ℕ in atTop, (N : ℝ)^r ≤ cutoff s N := by have ho := (isLittleO_log_rpow_atTop (sub_pos.mpr hrs)).bound (show (0:ℝ)<1 by norm_num) have hn : Tendsto (fun N : ℕ => (N : ℝ)) atTop atTop := tendsto_natCast_atTop_atTop filter_upwards [hn.eventually ho,hn.eventually (eventually_gt_atTop 1)] with N hN hp have hlog : 0 < Real.log N := Real.log_pos hp have hpow : 0 ≤ (N : ℝ)^(s-r) := Real.rpow_nonneg (by linarith) _ simp only [Real.norm_eq_abs,abs_of_nonneg hlog.le,abs_of_nonneg hpow,one_mul] at hN unfold cutoff apply (le_div_iff₀ hlog).mpr calc (N : ℝ)^r*Real.log N ≤ (N : ℝ)^r*(N : ℝ)^(s-r) := mul_le_mul_of_nonneg_left hN (Real.rpow_nonneg (by linarith) _) _ = (N : ℝ)^s := by rw [← Real.rpow_add (by linarith : (0:ℝ) (N : ℝ)) atTop atTop := tendsto_natCast_atTop_atTop have hl := Real.tendsto_log_atTop.comp hn filter_upwards [hl.eventually (eventually_gt_atTop (max 1 (8/η))), hn.eventually (eventually_gt_atTop 1)] with N hlog hN have hlog1 : 1 < Real.log N := (le_max_left _ _).trans_lt hlog have hlogη : 8/η < Real.log N := (le_max_right _ _).trans_lt hlog have hp : 0 < (N : ℝ)^s := Real.rpow_pos_of_pos (by linarith) _ have hi : (1:ℝ)/Real.log N ≤ η/8 := by apply (div_le_iff₀ (by linarith : 0 < Real.log N)).mpr have hh : 8 < Real.log N*η := (div_lt_iff₀ hη).mp hlogη nlinarith refine ⟨div_nonneg hp.le (by linarith),?_,?_⟩ · exact (div_lt_iff₀ (by linarith : 0 < Real.log N)).mpr (by nlinarith) · unfold cutoff calc (N : ℝ)^s/Real.log N = (1/Real.log N)*(N : ℝ)^s := by ring _ ≤ (η/8)*(N : ℝ)^s := mul_le_mul_of_nonneg_right hi hp.le lemma continuous_left_tail {s γ ε : ℝ} (hγ : γ < s) (hε : 0 < ε) (hs : ContinuousAt Theta s) : ∃ r : ℝ, γ < r ∧ r < s ∧ ∃ m₁ : ℕ, ∀ m : ℕ, m₁ ≤ m → F m r ≤ Theta s+ε := by obtain ⟨δ,hδ,hcont⟩ := Metric.continuousAt_iff.mp hs (ε/2) (by linarith) obtain ⟨r,hrlo,hrhi⟩ := exists_between (show max γ (s-δ) < s by exact max_lt hγ (by linarith)) have hγr : γ < r := (le_max_left _ _).trans_lt hrlo have hdist : dist r s < δ := by rw [Real.dist_eq,abs_of_neg (sub_neg.mpr hrhi)] have hh := (le_max_right γ (s-δ)).trans_lt hrlo linarith have ht : Theta r < Theta s+ε := by have hh := (abs_lt.mp (show |Theta r-Theta s| < ε/2 from by simpa only [Real.dist_eq] using hcont hdist)).2 linarith obtain ⟨m₁,hm⟩ := eventually_atTop.mp (eventually_lt_of_limsup_lt ht (F_bdd r)) exact ⟨r,hγr,hrhi,m₁,fun m hm₁ => (hm m hm₁).le⟩ lemma half_boundary (N : ℕ) (hN : 4 ≤ N) : let m₀ := (N+1)/2 2 ≤ m₀ ∧ m₀ < N ∧ (m₀ : ℝ)*((m₀ : ℝ)-1) ≤ (N : ℝ)*((N : ℝ)-1)/2 := by dsimp have hm : 2 ≤ (N+1)/2 := by omega have hlt : (N+1)/2 < N := by omega refine ⟨hm,hlt,?_⟩ have h2 : 2*((N+1)/2) ≤ N+1 := by omega have h2R : (2:ℝ)*((N+1)/2 : ℕ) ≤ (N : ℝ)+1 := by exact_mod_cast h2 have hNR : (4:ℝ) ≤ N := by exact_mod_cast hN have hmR : (2:ℝ) ≤ ((N+1)/2 : ℕ) := by exact_mod_cast hm nlinarith /-- Actual core configurations at arbitrarily small density error and arbitrarily large cardinality. No field or position assumptions are introduced. -/ theorem arbitrarily_large_core {s γ : ℝ} (hγ : 0 < γ) (hγs : γ < s) (hs : ContinuousAt Theta s) (η : ℝ) (hη : 0 < η) (M : ℕ) : ∃ (P : Finset Plane) (C : Finset P), M < C.card ∧ (P.card+1)/2 < C.card ∧ (C.card : ℝ)^γ ≤ cutoff s P.card ∧ (Theta s-η)*(C.card : ℝ)*((C.card : ℝ)-1) ≤ ((planarFibers P s).edgeSet (cutoff s P.card) C).card ∧ (((planarFibers P s).edgeSet (cutoff s P.card) C).card : ℝ) ≤ (Theta s+η/8)*(C.card : ℝ)*((C.card : ℝ)-1) ∧ (∀ x ∈ C, 2*(Theta s-η)*((C.card : ℝ)-1) ≤ (planarFibers P s).degree (cutoff s P.card) C x) := by obtain ⟨r,hγr,hrs,m₁,hm₁⟩ := continuous_left_tail hγs (show 0 < η/8 by positivity) hs obtain ⟨N,hN,hF⟩ := extremal_subsequence s have hlarge : ∀ᶠ i in atTop, 4+2*M+2*m₁ ≤ N i := hN.eventually (eventually_ge_atTop _) have happrox : ∀ᶠ i in atTop, Theta s-η/8 ≤ F (N i) s := (hF.eventually_const_lt (by linarith : Theta s-η/8 < Theta s)).mono (fun _ h => h.le) obtain ⟨i,hi,hcut,hpow,hinit⟩ := (hlarge.and ((hN.eventually (eventual_cutoff_loss s η hη)).and ((hN.eventually (eventual_power_below_cutoff hrs)).and happrox))).exists obtain ⟨P,hP,hPF⟩ := F_attained (N i) (by omega) s have hn : 4 ≤ P.card := by omega obtain ⟨hm0,hm,hbd⟩ := half_boundary P.card hn have hm1 : m₁ ≤ (P.card+1)/2 := by omega have hη8 := hcut.2.2 have hpowP : (P.card : ℝ)^r ≤ cutoff s P.card := by simpa [hP] using hpow have hthreshold : ∀ m : ℕ, (P.card+1)/2 ≤ m → m ≤ P.card → (m : ℝ)^r ≤ cutoff s P.card := by intro m _ hmP exact (Real.rpow_le_rpow (Nat.cast_nonneg m) (by exact_mod_cast hmP) (by linarith)).trans hpowP obtain ⟨C,hC,hlo,hhi,hdeg⟩ := finite_extremal_core P s r (cutoff s P.card) (Theta s) η ((P.card+1)/2) (by omega) hm0 hm hη (by simpa [hP] using hcut.1) (by simpa [hP] using hcut.2.1) (by simpa [hP] using hη8) (by simpa [hPF] using hinit) hbd hthreshold (fun m hm _ => hm₁ m (hm1.trans hm)) have hCn : C.card ≤ P.card := by simpa using Finset.card_le_univ C have hCpos : (1:ℝ) ≤ C.card := by exact_mod_cast (show 1 ≤ C.card by omega) refine ⟨P,C,by omega,hC,?_,hlo,hhi,hdeg⟩ exact (Real.rpow_le_rpow_of_exponent_le hCpos hγr.le).trans (hthreshold C.card hC.le hCn) end WeakPinned namespace WeakPinned open scoped BigOperators lemma restricted_size_lt_card (P : Finset Plane) (s : ℝ) (D : Finset P) (e : fiberIndices P s) (hs : (planarFibers P s).source e ∈ D) : (planarFibers P s).size D e < D.card := by have hh : (planarFibers P s).targets e ∩ D ⊆ D.erase ((planarFibers P s).source e) := by intro y hy obtain ⟨hyT,hyD⟩ := Finset.mem_inter.mp hy refine Finset.mem_erase.mpr ⟨?_,hyD⟩ intro he subst y exact (planarFibers P s).no_loop e hyT have hc := Finset.card_le_card hh rw [Finset.card_erase_of_mem hs] at hc have hp := Finset.card_pos.mpr ⟨_,hs⟩ change (planarFibers P s).size D e ≤ D.card-1 at hc omega lemma heavySubset_zero (P : Finset Plane) (s K lam : ℝ) (C D : Finset P) (hlam : 0 ≤ lam) (hsmall : (D.card : ℝ) ≤ lam*K) : (∑ e, if heavyFiber P s K lam C D e ∧ (planarFibers P s).source e ∈ D then ((planarFibers P s).size D e : ℝ) else 0) = 0 := by apply Finset.sum_eq_zero intro e _ rw [ite_eq_right] rintro ⟨⟨ha,hh⟩,hs⟩ have hk := mul_le_mul_of_nonneg_left (le_of_lt ha.2) hlam have hm : ((planarFibers P s).size D e : ℝ) < D.card := by exact_mod_cast restricted_size_lt_card P s D e hs linarith /-- A uniform extremal bound valid even for tiny subsets: their contribution is zero, rather than an uncontrolled diagonal error. -/ theorem heavySubset_uniform (P : Finset Plane) (s K lam r B : ℝ) (C D : Finset P) (hlam : 0 ≤ lam) (hB : 0 ≤ B) (m₁ : ℕ) (hm1 : 2 ≤ m₁) (hT : (m₁ : ℝ) ≤ lam*K) (hthreshold : (D.card : ℝ)^r ≤ lam*K) (hext : ∀ m : ℕ, m₁ ≤ m → m ≤ P.card → F m r ≤ B) : (∑ e, if heavyFiber P s K lam C D e ∧ (planarFibers P s).source e ∈ D then ((planarFibers P s).size D e : ℝ) else 0) ≤ B*(D.card : ℝ)*((D.card : ℝ)-1) := by by_cases hm : D.card < m₁ · have hDm : (D.card : ℝ) ≤ m₁ := by exact_mod_cast hm.le rw [heavySubset_zero P s K lam C D hlam (hDm.trans hT)] by_cases hzero : D.card=0 · simp [hzero] · have hh : (1:ℝ) ≤ D.card := by exact_mod_cast (show 1 ≤ D.card by omega) positivity · have hmge : m₁ ≤ D.card := by omega have hDn : D.card ≤ P.card := by simpa using Finset.card_le_univ D have hh : (2:ℝ) ≤ D.card := by exact_mod_cast hm1.trans hmge have hp : 0 < (D.card : ℝ)*((D.card : ℝ)-1) := by nlinarith have hf := fraction_le_F (planeSubset P D) r (by simpa [planeSubset_card] using hm1.trans hmge) rw [pairFraction,planeSubset_card,div_le_iff₀ hp] at hf have hc := heavySubset_count P s K lam r C D hlam hthreshold have he := mul_le_mul_of_nonneg_right (hext D.card hmge hDn) hp.le nlinarith lemma eventual_power_below_logsquare {s r : ℝ} (hrs : r < s) : ∀ᶠ N : ℕ in Filter.atTop, (N : ℝ)^r ≤ cutoff s N / Real.log N := by have ho := (isLittleO_log_rpow_rpow_atTop (2:ℝ) (sub_pos.mpr hrs)).bound (show (0:ℝ)<1 by norm_num) have hn : Filter.Tendsto (fun N : ℕ => (N : ℝ)) Filter.atTop Filter.atTop := tendsto_natCast_atTop_atTop filter_upwards [hn.eventually ho,hn.eventually (Filter.eventually_gt_atTop 1)] with N hN hp have hlog : 0 < Real.log N := Real.log_pos hp have hpow : 0 ≤ (N : ℝ)^(s-r) := Real.rpow_nonneg (by linarith) _ simp only [Real.norm_eq_abs,Real.rpow_two,one_mul,abs_of_nonneg (sq_nonneg (Real.log (N : ℝ))),abs_of_nonneg hpow] at hN unfold cutoff rw [div_div] apply (le_div_iff₀ (mul_pos hlog hlog)).mpr calc (N : ℝ)^r*(Real.log N*Real.log N) ≤ (N : ℝ)^r*(N : ℝ)^(s-r) := by apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (by linarith) _) nlinarith _ = (N : ℝ)^s := by rw [← Real.rpow_add (by linarith : (0:ℝ) hm₁ j (hmr.trans hj)) · intro D hD have hDP : (D.card : ℝ)^t ≤ (P.card : ℝ)^((1-c)*t) := by calc (D.card : ℝ)^t ≤ ((C.card : ℝ)^(1-c))^t := Real.rpow_le_rpow (Nat.cast_nonneg _) hD ht0.le _ = (C.card : ℝ)^((1-c)*t) := (Real.rpow_mul (Nat.cast_nonneg _) _ _).symm _ ≤ (P.card : ℝ)^((1-c)*t) := Real.rpow_le_rpow (Nat.cast_nonneg _) (by exact_mod_cast hCn) (by positivity) exact heavySubset_uniform P s (cutoff s P.card) (1/Real.log C.card) t η C D hlam hη.le m hm2 hT (hDP.trans (hpt.trans hcomp)) (fun j hj _ => (hm₂ j (hmt.trans hj)).le) end WeakPinned end end end end end end end end end OAI